1. Psy 8960, Fall ‘06 Introduction to MRI1 Fourier transforms 1D: square wave 2D: k x and k y 2D: FOV and resolution 2D: spike artifacts 3D

2. Psy 8960, Fall ‘06 Introduction to MRI2 Fourier (de)composition of a square wave Fundamental frequency: Fundamental + 1 st harmonic: Fundamental + 2 harmonics: Fundamental + 3 harmonics:

3. Psy 8960, Fall ‘06 Introduction to MRI3 Fourier (de)composition of a square wave 16s

4. Psy 8960, Fall ‘06 Introduction to MRI4 The 0 th Fourier component is the mean (DC)

5. Psy 8960, Fall ‘06 Introduction to MRI5 Even symmetry = lack of imaginary component in transform

6. Psy 8960, Fall ‘06 Introduction to MRI6 A real image should have symmetric k- space

7. Psy 8960, Fall ‘06 Introduction to MRI7 secondscycles per second Discrete Fourier transform: the effect of sampling rate

8. Psy 8960, Fall ‘06 Introduction to MRI8 Discrete Fourier transform: the effect of sampling window secondscycles per second

9. Psy 8960, Fall ‘06 Introduction to MRI9 Fourier relationships Big step size in one domain = small FOV in the other Large extent (FOV) in one domain = small step size in the other Multiplication in one domain = convolution in the other Symmetry in one domain = no imaginary part in the other

10. Psy 8960, Fall ‘06 Introduction to MRI10 Time domainFrequency domain secondscycles per second real imag

11. Psy 8960, Fall ‘06 Introduction to MRI11 Time domainFrequency domain secondscycles per second real imag

12. Psy 8960, Fall ‘06 Introduction to MRI12 Time domainFrequency domain secondscycles per second

13. Psy 8960, Fall ‘06 Introduction to MRI13 Time domainFrequency domain secondscycles per second

14. Psy 8960, Fall ‘06 Introduction to MRI14 Time domainFrequency domain secondscycles per second

15. Psy 8960, Fall ‘06 Introduction to MRI15 original imagefiltered with gaussian filterfiltered with hard filter